arXiv subjects

Maurice Pouzet

Publications and source records attributed to Maurice Pouzet.

At least 37 records · Page 2Linked to original sources

arXiv2018

A syntactic approach to the MacNeille completion of $\boldΛ^{\ast}$, the free monoid over an ordered alphabet $\bold Λ$

Let $Λ^{\ast}$ be the free monoid of (finite) words over a not necessarily finite alphabet $Λ$, which is equipped with some (partial) order. This ordering lifts to $Λ^{\ast}$, where it extends the divisibility ordering of words. The MacNeille completion of $Λ^{\ast}$ constitutes a complete lattice ordered monoid and is realized by the system of "closed" lower sets in $Λ^*$ (ordered by inclusion) or its isomorphic copy formed of the "closed" upper sets (ordered by reverse inclusion). Under some additional hypothesis on $Λ$, one can easily identify the closed lower sets as the finitely generated ones, whereas it is more complicated to determine the closed upper sets. For a fairly large class of ordered sets $Λ$ (including complete lattices as well as antichains) one can generate the closure of any upper set of words by means of binary operations ( "syntactic rules") thus obtaining an efficient procedure to test closedness. Closed upper set of words are involved in an embedding theorem for valuated oriented graphs. In fact, generalized paths (so-called "zigzags") are encoded by words over an alphabet $Λ$. Then the valuated oriented graphs which are "isometrically" embeddable in a product of zigzags have the characteristic property that the words corresponding to the zigzags between any pair of vertices form a closed upper set in $Λ$.

math.CO↗

arXiv2018

A fixed point theorem for commuting families of relational homomorphisms. Application to metric spaces, oriented graphs and ordered sets

We extend to binary relational systems the notion of compact and normal structure, introduced by J.P.Penot for metric spaces, and we prove that for the involutive and reflexive ones, every commuting family of relational homomorphisms has a common fixed point. The proof is based upon the clever argument that J.B.Baillon discovered in order to show that a similar conclusion holds for bounded hyperconvex metric spaces and then refined by the first author to metric spaces with a compact and normal structure. Since the non-expansive mappings are relational homomorphisms, our result includes those of T.C.Lim, J.B.Baillon and the first author. We show that it extends the Tarski's fixed point theorem to graphs which are retracts of reflexive oriented zigzags of bounded length. Doing so, we illustrate the fact that the consideration of binary relational systems or of generalized metric spaces are equivalent.

math.FA↗

arXiv2018

Interval orders, semiorders and ordered groups

We prove that the order of an ordered group is an interval order if and only if it is a semiorder. Next, we prove that every semiorder is isomorphic to a collection $\mathcal J$ of intervals of some totally ordered abelian group, these intervals being of the form $[x, x+ α[$ for some positive $α$. We describe ordered groups such that the ordering is a semiorder and we introduce threshold groups generalizing totally ordered groups. We show that the free group on finitely many generators and the Thompson group $\mathbb F$ can be equipped with a compatible semiorder which is not a weak order. On another hand, a group introduced by Clifford cannot.

math.CO↗

arXiv2018

Some relational structures with polynomial growth and their associated algebras II: Finite generation

The profile of a relational structure $R$ is the function $φ_R$ which counts for every integer $n$ the number, possibly infinite, $φ_R(n)$ of substructures of $R$ induced on the $n$-element subsets, isomorphic substructures being identified. If $φ_R$ takes only finite values, this is the Hilbert function of a graded algebra associated with $R$, the age algebra $A(R)$, introduced by P.~J.~Cameron. In a previous paper, we studied the relationship between the properties of a relational structure and those of their algebra, particularly when the relational structure $R$ admits a finite monomorphic decomposition. This setting still encompasses well-studied graded commutative algebras like invariant rings of finite permutation groups, or the rings of quasi-symmetric polynomials. In this paper, we investigate how far the well know algebraic properties of those rings extend to age algebras. The main result is a combinatorial characterization of when the age algebra is finitely generated. In the special case of tournaments, we show that the age algebra is finitely generated if and only if the profile is bounded. We explore the Cohen-Macaulay property in the special case of invariants of permutation groupoids. Finally, we exhibit sufficient conditions on the relational structure that make naturally the age algebra into a Hopf algebra.

math.CO↗

arXiv2017

Jónsson posets

According to Kearnes and Oman (2013), an ordered set $P$ is \emph{Jónsson} if it is infinite and the cardinality of every proper initial segment of $P$ is strictly less than the cardinaliy of $P$. We examine the structure of Jónsson posets.

math.LO↗

arXiv2017

Operations preserving equivalence relations

In 2014, Cégielski, Grigorieff and Guessarian characterized unary self-maps on the set $\mathbb{Z}$ of integers which preserve all congruences of the additive group. In this note, we propose a shorter and straigthforward proof. We replace this result in the frame of universal algebra.

math.CO↗

arXiv2017

Free monoids and generalized metric spaces

Let $A$ be an ordered alphabet, $A^{\ast}$ be the free monoid over $A$ ordered by the Higman ordering, and let $F(A^{\ast})$ be the set of final segments of $A^{\ast}$. With the operation of concatenation, this set is a monoid. We show that the submonoid $F^{\circ}(A^{\ast}):= F(A^{\ast})\setminus \{\emptyset\}$ is free. The MacNeille completion $N(A^{\ast})$ of $A^{\ast}$ is a submonoid of $F(A^{\ast})$. As a corollary, we obtain that the monoid $N^{\circ}(A^{\ast}):=N(A^{\ast})\setminus \{\emptyset\}$ is free. We give an interpretation of the freeness of $F^{\circ}(A^{\ast})$ in the category of metric spaces over the Heyting algebra $V:= F(A^{\ast})$, with the non-expansive mappings as morphisms. Each final segment of $A^{\ast}$ yields the injective envelope $\mathcal S_F$ of a two-element metric space over $V$. The uniqueness of the decomposition of $F$ is due to the uniqueness of the block decomposition of the graph $\mathcal {G}_{F}$ associated to this injective envelope.

math.CO↗

arXiv2016

Invariant subsets of scattered trees. An application to the tree alternative property of Bonato and Tardif

A tree is scattered if no subdivision of the complete binary tree is a subtree. Building on results of Halin, Polat and Sabidussi, we identify four types of subtrees of a scattered tree and a function of the tree into the integers at least one of which is preserved by every embedding. With this result and a result of Tyomkyn, we prove that the tree alternative property conjecture of Bonato and Tardif holds for scattered trees and a conjecture of Tyomkin holds for locally finite scattered trees.

math.CO↗

arXiv2015

Length of an intersection

A poset $\bfp$ is well-partially ordered (WPO) if all its linear extensions are well orders~; the supremum of ordered types of these linear extensions is the {\em length}, $\ell(\bfp)$ of $\bfp$. We prove that if the vertex set $X$ of $\bfp$ is infinite, of cardinality $κ$, and the ordering $\leq$ is the intersection of finitely many partial orderings $\leq_i$ on $X$, $1\leq i\leq n$, then, letting $\ell(X,\leq_i)=κ\multordby q_i+r_i$, with $r_i<κ$, denote the euclidian division by $κ$ (seen as an initial ordinal) of the length of the corresponding poset~:\[ \ell(\bfp)< κ\multordby\bigotimes_{1\leq i\leq n}q_i+ \Big|\sum_{1\leq i\leq n} r_i\Big|^+ \] where $|\sum r_i|^+$ denotes the least initial ordinal greater than the ordinal $\sum r_i$. This inequality is optimal (for $n\geq 2$).

math.LO↗

arXiv2015

Semirigid systems of three equivalence relations

A system $\mathcal M$ of equivalence relations on a set $E$ is \emph{semirigid} if only the identity and constant functions preserve all members of $\mathcal M$. We construct semirigid systems of three equivalence relations. Our construction leads to the examples given by Zádori in 1983 and to many others and also extends to some infinite cardinalities. As a consequence, we show that on every set of at most continuum cardinality distinct from $2$ and $4$ there exists a semirigid system of three equivalence relations.

math.CO↗

arXiv2015

On scattered convex geometries

A convex geometry is a closure space satisfying the anti-exchange axiom. For several types of algebraic convex geometries we describe when the collection of closed sets is order scattered, in terms of obstructions to the semilattice of compact elements. In particular, a semilattice $Ω(η)$, that does not appear among minimal obstructions to order-scattered algebraic modular lattices, plays a prominent role in convex geometries case. The connection to topological scatteredness is established in convex geometries of relatively convex sets.

math.CO↗

arXiv2015

Hereditarily rigid relations

An $h$-ary relation $\r$ on a finite set $A$ is said to be \emph{hereditarily rigid} if the unary partial functions on $A$ that preserve $\r$ are the subfunctions of the identity map or of constant maps. A family of relations ${\mathcal F}$ is said to be \emph{hereditarily strongly rigid} if the partial functions on $A$ that preserve every $\r \in {\mathcal F}$ are the subfunctions of projections or constant functions. In this paper we show that hereditarily rigid relations exist and we give a lower bound on their arities. We also prove that no finite hereditarily strongly rigid families of relations exist and we also construct an infinite hereditarily strongly rigid family of relations.

math.CO↗

arXiv2015

Isomorphy up to complementation

Considering uniform hypergraphs, we prove that for every non-negative integer $h$ there exist two non-negative integers $k$ and $t$ with $k\leq t$ such that two $h$-uniform hypergraphs ${\mathcal H}$ and ${\mathcal H}'$ on the same set $V$ of vertices, with $| V| \geq t$, are equal up to complementation whenever ${\mathcal H}$ and ${\mathcal H}'$ are $k$-{hypomorphic up to complementation}. Let $s(h)$ be the least integer $k$ such that the conclusion above holds and let $v(h)$ be the least $t$ corresponding to $k=s(h)$. We prove that $s(h)= h+2^{\lfloor \log_2 h\rfloor} $. In the special case $h=2^{\ell}$ or $h=2^{\ell}+1$, we prove that $v(h)\leq s(h)+h$. The values $s(2)=4$ and $v(2)=6$ were obtained in a previous work.

math.CO↗

arXiv2014

Décomposition monomorphe des structures relationnelles et profil de classes héréditaires

We present a structural approach of some results about jumps in the behavior of the profile (alias generating function) of hereditary classes of finite structures. We start with the following notion due to N.Thiéry and the second author. A \emph{monomorphic decomposition} of a relational structure $R$ is a partition of its domain $V(R)$ into a family of sets $(V_x)_{x\in X}$ such that the restrictions of $R$ to two finite subsets $A$ and $A'$ of $V(R)$ are isomorphic provided that the traces $A\cap V_x$ and $A'\cap V_x$ have the same size for each $x\in X$. Let $\mathscr S_μ$ be the class of relational structures of signature $μ$ which do not have a finite monomorphic decomposition. We show that if a hereditary subclass $\mathscr D$ of $\mathscr S_μ$ is made of ordered relational structures then it contains a finite subset $\mathfrak A$ such that every member of $\mathscr D$ embeds some member of $\mathfrak A$. Furthermore, for each $R\in \mathfrak A$ the profile of the age $\mathcal A(R)$ of $R$ (made of finite substructures of $R$) is at least exponential. We deduce that if the profile of a hereditary class of finite ordered structures is not bounded by a polynomial then it is at least exponential. This result is a part of classification obtained by Balogh, Bollobás and Morris (2006) for ordered graphs. {\it To cite this article: Djamila Oudrar, Maurice Pouzet, C. R. Acad. Sci. Paris, Ser. I.}

math.CO↗

arXiv2014

Profile and hereditary classes of ordered relational structures

Let $\mathfrak{C}$ be a class of finite combinatorial structures. The \textit{profile} of $\mathfrak{C}$ is the function $φ_{\mathfrak{C}}$ which counts, for every integer $n$, the number $φ_{\mathfrak{C}}(n)$ of members of $\mathfrak{C}$ defined on $n$ elements, isomorphic structures been identified. The \textit{generating function of} $\mathfrak{C}$ is $\mathcal {H}_{\mathfrak{C}}(x):=\sum_{n\geqq 0}φ_{\mathfrak{C}}(n)x^{n}$. Many results about the behavior of the function $φ_{\mathfrak{C}}$ have been obtained. Albert and Atkinson have shown that the generating series of several classes of permutations are algebraic. In this paper, we show how their results extend to classes of ordered binary relational structures; putting emphasis on the notion of hereditary well quasi order, we discuss some of their questions and answer one.

math.CO↗

arXiv2014

Pairs of orthogonal countable ordinals

We characterize pairs of orthogonal countable ordinals. Two ordinals $α$ and $β$ are orthogonal if there are two linear orders $A$ and $B$ on the same set $V$ with order types $α$ and $β$ respectively such that the only maps preserving both orders are the constant maps and the identity map. We prove that if $α$ and $β$ are two countable ordinals, with $α\leq β$, then $α$ and $β$ are orthogonal if and only if either $ω+ 1\leq α$ or $α=ω$ and $β< ωβ$.

math.CO↗

arXiv2014

Some relational structures with polynomial growth and their associated algebras I: Quasi-polynomiality of the profile

The profile of a relational structure $R$ is the function $φ_R$ which counts for every integer $n$ the number $φ_R(n)$, possibly infinite, of substructures of $R$ induced on the $n$-element subsets, isomorphic substructures being identified. If $φ_R$ takes only finite values, this is the Hilbert function of a graded algebra associated with $R$, the age algebra introduced by P. J. Cameron. In this paper we give a closer look at this association, particularly when the relational structure $R$ admits a finite monomorphic decomposition. This setting still encompass well-studied graded commutative algebras like invariant rings of finite permutation groups, or the rings of quasi-symmetric polynomials. We prove that $φ_R$ is eventually a quasi-polynomial, this supporting the conjecture that, under mild assumptions on $R$, $φ_R$ is eventually a quasi-polynomial when it is bounded by some polynomial.

math.CO↗

arXiv2013

Claw-freeness, 3-homogeneous subsets of a graph and a reconstruction problem

We describe ${\rm Forb}\{K_{1,3}, \bar {K_{1,3}}\}$, the class of graphs $G$ such that $G$ and its complement $\bar{G}$ are claw-free. With few exceptions, it is made of graphs whose connected components consist of cycles of length at least 4, paths, and of the complements of these graphs. Considering the hypergraph ${\mathcal H}^{(3)}(G)$ made of the 3-element subsets of the vertex set of a graph $G$ on which $G$ induces a clique or an independent subset, we deduce from above a description of the Boolean sum $G\dot{+}G'$ of two graphs $G$ and $G'$ giving the same hypergraph. We indicate the role of this latter description in a reconstruction problem of graphs up to complementation.

math.CO↗